# On the Thermal Capacity of Solids

## Abstract

**:**

## 1. Introduction

#### 1.1. Energy and Entropy

#### 1.2. Outline

## 2. Materials and Methods

## 3. Entropy Capacity

## 4. Entropy Capacity versus “Heat Capacity”

## 5. Analogy: Storage of a Fluid in a Vessel

## 6. Entropy Capacity of Diamond and Graphite

## 7. Reaction Entropy

## 8. Caloric Materials

## 9. Thermoelectrics and Thermal Conductivity

## 10. Phononic Contributions to Entropy Capacity: Debye Model

## 11. Phononic and Electronic Contributions to Entropy Capacity

## 12. Discussion

#### 12.1. Thermal Capacity

#### 12.2. Units of Entropy and Entropy Capacity

#### 12.3. Confusion and Resolution

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Symbols

${\tilde{C}}_{\u2107}$ | specific temperature coefficient of energy (at a constant electrical field) (specific “heat capacity” at a constant electrical field) |

${C}_{V}$ | temperature coefficient of energy (at constant volume) (“heat capacity” at constant volume) |

${\widehat{C}}_{V}$ | molar temperature coefficient of energy (molar “heat capacity” at constant volume) |

${C}_{V,N}$ | temperature coefficient of energy at a constant number of particles (“heat capacity” at constant volume and at a constant number of particles) |

${C}_{p}$ | temperature coefficient of enthalpy (“heat capacity” at constant pressure) |

${\widehat{C}}_{p}$ | molar temperature coefficient of enthalpy (molar “heat capacity” at constant pressure) |

${\tilde{C}}_{p}$ | specific temperature coefficient of enthalpy (specific isobaric “heat capacity”) |

${C}_{p,N}$ | temperature coefficient of enthalpy at a constant number of particles (“heat capacity” at constant pressure and at a constant number of particles) |

${D}_{\mathrm{th}}$ | thermal diffusivity (diffusion coefficient of “heat”, diffusion coefficient of entropy) |

$\mathcal{D}\left({E}_{\mathrm{F}}\right)$ | electronic density of states at the Fermi energy |

E | energy |

${E}_{\mathrm{F}}$ | Fermi energy |

$\u2107$ | electrical field |

f | dimensionless thermoelectric figure of merit (see also $zT$) |

H | enthalpy |

${k}_{\mathrm{B}}$ | Boltzmann’s constant |

${K}_{\u2107}$ | entropy capacity at constant electrical field |

${\tilde{K}}_{\u2107}$ | specific entropy capacity at constant electrical field |

${K}_{\mathcal{H}}$ | entropy capacity at constant magnetic field |

${K}_{M}$ | entropy capacity at constant magnetisation |

${K}_{p}$ | entropy capacity at constant pressure (isobaric entropy capacity) |

${\widehat{K}}_{p}$ | molar isobaric entropy capacity |

${\widehat{K}}_{p,i}$ | molar isobaric entropy capacity of substance i |

${\widehat{K}}_{p,j}$ | molar isobaric entropy capacity of substance j |

${K}_{p,N}$ | entropy capacity at constant pressure and at a constant number of particles |

${\tilde{K}}_{p}$ | specific isobaric entropy capacity |

${K}_{P}$ | entropy capacity at constant (electrical) polarisation |

${K}_{V}$ | entropy capacity at constant volume (isochoric entropy capacity) |

${K}_{V,N}$ | entropy capacity at constant volume and at a constant number of particles |

${\widehat{K}}_{V}$ | molar isochoric entropy capacity |

${K}_{\sigma}$ | entropy capacity at constant stress |

${K}_{\epsilon}$ | entropy capacity at constant strain |

N | amount of substance (number of particles), given in mol |

p | pressure |

P | (electrical) polarisation |

R | universal gas constant |

S | entropy |

T | absolute temperature |

V | volume |

x | integration variable in Debye model |

$zT$ | dimensionless thermoelectric figure of merit (see also f) |

$\gamma $ | molar isochoric entropy capacity of the electron gas (Sommerfeld coefficient) |

$\beta $ | factor in the Debye model (low-temperature limit) |

$\mathsf{\Delta}S$ | reaction entropy |

$\mathsf{\Delta}\widehat{S}$ | molar reaction entropy |

$\mathsf{\Delta}T$ | temperature difference |

$\epsilon $ | strain |

$\lambda $ | open-circuited specific “heat” conductivity |

$\mathsf{\Lambda}$ | open-circuited specific entropy conductivity |

${\nu}_{i}$ | stoichiometric coefficient of substance i |

${\nu}_{j}$ | stoichiometric coefficient of substance j |

$\rho $ | density |

$\sigma $ | (mechanical) stress |

${\mathsf{\Theta}}_{\mathrm{D}}$ | Debye temperature |

## Appendix A. Entropy Capacity and “Heat Capacity” of Graphite and Diamond

#### Appendix A.1. “Heat Capacity” of Graphite and Diamond According to Vassiliev and Taldrik

**Figure A1.**Temperature dependence of the molar temperature coefficient of enthalpy ${\widehat{C}}_{p}$ of graphite and diamond and the molar temperature coefficient of energy ${\widehat{C}}_{V}$ according to the Dulong–Petit relationship (constant) of the classical ideal gas. ${\widehat{C}}_{p}$ values were calculated according to the multiparameter model of Vassiliev and Taldrik [35] (see Appendix A.3).

#### Appendix A.2. Entropy Capacity and “Heat Capacity” of Diamond

**Figure A2.**(

**a**) Molar entropy capacity of diamond (${\widehat{K}}_{p}$ ) according to the multiparameter model of Vassiliev and Taldrik [35] (see Appendix A.3), compared to the Debye model (${\widehat{K}}_{V}$) with two different Debye temperatures (1830 K [11] and 2240 K [38]). The low-temperature Debye model and Dulong–Petit relationship are also displayed. (

**b**) Molar temperature coefficient of enthalpy ${\widehat{C}}_{p}$ of diamond according to the multiparameter model by Vassiliev and Taldrik [35] (see Appendix A.3), compared to the Debye model (${\widehat{C}}_{V}$) with two different Debye temperatures (1830 K [11] and 2240 K [38]). The low-temperature Debye model and Dulong–Petit relationship are also displayed.

#### Appendix A.3. Multiparameter Modelling of the Entropy Capacity and “Heat Capacity” of Graphite and Diamond

**Table A1.**Parameters for Equations (A1)–(A5) according to the Debye–Maier–Kelley hybrid model in the range of 0.1 K to the melting point for diamond (Table 6, 1b, in [35]) and graphite (Table 6, 2c, in [35]). Reprinted from Journal of Alloys and Compounds, 872, Vassiliev, V.P., Taldrik, A.F., Description of the heat capacity of solid phases by a multiparameter family of functions, 159682, Copyright (2021), with permission from Elsevier.

Phase | ${\mathit{T}}_{0}$ | ${\mathit{A}}_{1}$ | ${\mathsf{\Theta}}_{\mathsf{D},1}$ | ${\mathit{A}}_{2}$ | ${\mathsf{\Theta}}_{\mathsf{D},2}$ | ${\mathit{A}}_{3}$ | ${\mathsf{\Theta}}_{\mathsf{D},3}$ | a | b |
---|---|---|---|---|---|---|---|---|---|

Diamond | 1366 | 0.031 | 1833.6 | 0.488 | 1968.7 | 0.482 | 1824.5 | 24.59 | 0.287 |

Graphite | 282.6 | 0.773 | 1949.9 | 0.114 | 426.4 | 0.114 | 947.9 | 24.25 | 0.848 |

#### Appendix A.4. Comparison to Wiberg’s Book

**Table A2.**Amount of accumulated entropy ${S}^{\mathsf{\Delta}T}$ in graphite and diamond in equal temperature intervals $\mathsf{\Delta}T$ of 300 K and the associated integrated reaction entropy $\mathsf{\Delta}{S}^{\mathsf{\Delta}T}$.

Temperature Interval | ${\mathit{S}}_{\mathbf{graphite}}^{\mathsf{\Delta}\mathit{T}}$ | ${\mathit{S}}_{\mathbf{diamond}}^{\mathsf{\Delta}\mathit{T}}$ | $\mathsf{\Delta}{\mathit{S}}^{\mathsf{\Delta}\mathit{T}}$ | |||
---|---|---|---|---|---|---|

Wiberg [10]
^{1,2} | This work
^{3} | Wiberg [10]
^{1,2} | This Work
^{3} | Wiberg [10]
^{1,2} | This Work
^{3} | |

1500–1800 | N/A | 4.43 | N/A | 4.29 | N/A | 0.14 |

1200–1500 | N/A | 5.22 | N/A | 5.07 | N/A | 0.15 |

900–1200 | 6.28 | 6.32 | 6.28 | 6.12 | 0 | 0.20 |

600–900 | 7.49 | 7.80 | 7.49 | 7.39 | 0 | 0.41 |

300–600 | 8.83 | 8.95 | 7.95 | 7.56 | 0.88 | 1.39 |

0–300 | 5.78 | 5.66 | 2.43 | 2.33 | 3.35 | 3.33 |

^{1}Wiberg [10] presented values of entropy in the unit 1 Clausius = 1 cal · K

^{−1}= 4.1868 J · K

^{−1}.

^{2}Wiberg likely used thermochemical data (pp. 24, 149, [10]) from the Landolt–Börnstein [63] to construct entropy capacity versus temperature diagrams in units of Clausius per Kelvin versus Kelvin.

^{3}Thermochemical data for graphite and diamond used in this work rely on the multiparameter model of Vassiliev and Taldrik [35].

## Appendix B. Entropy Capacity and “Heat Capacity” of Barium Titanate

**Figure A3.**(

**a**) Graph of the specific entropy capacity ${\tilde{K}}_{\u2107}$ of BaTiO${}_{3}$ versus temperature for zero field and $\u2107$ = 10 kV cm${}^{-1}$; and (

**b**) Graph of the specific “heat capacity” ${\tilde{C}}_{\u2107}$ of BaTiO${}_{3}$ versus temperature for zero field and 4 different field strength levels from [52]. Figure (

**b**) was reprinted from Physica Status Solidi A, 209, Bai, Y., Ding, K., Zheng, G.P., Shi, S.Q., Qiao, L., Entropy-change measurement of electrocaloric effect of BaTiO${}_{3}$ single crystal., 941–944, Copyright (2012), with permission from Wiley-VCH.

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**Figure 1.**The capacity of a glass to store a fluid (here red wine) depends on the shape of the glass and changes with the fluid level. Depending on the shape of the glass vessel, different amounts of fluid are needed to raise the fluid level by 25 mm each. From left to right, the beakers contain 63 mL, 120 mL, 83 mL and 54 mL of fluid, which add to 320 mL when filled into the glass. A video sequence of filling the glass is available as Video S1.

**Figure 2.**Temperature dependence of isobaric entropy capacity ${K}_{p}$ of 1 mol carbon allotropes: (

**a**) graphite; (

**b**) diamond; (

**c**) graphite and diamond with differences highlighted. Entropy capacities were calculated according to the multiparameter model by Vassiliev and Taldrik [35] (see Appendix A.3). Following Wiberg [10].

**Figure 3.**Temperature dependence of molar isobaric entropy capacity ${\widehat{K}}_{p}$ of graphite and diamond and temperature dependence of molar isochoric entropy capacity ${\widehat{K}}_{V}$ according to the Dulong–Petit relationship (hyperbolic) of the classical ideal gas. Isobaric entropy capacities were calculated according to the multiparameter model by Vassiliev and Taldrik [35] (see Appendix A.3). Following Wiberg [10].

**Figure 4.**Molar reaction entropy of the transformation of diamond into graphite versus temperature as calculated according to the multiparameter model by Vassiliev and Taldrik [35] (see Appendix A.3). Following Wiberg [10].

**Figure 5.**Examples of entropy capacity K of caloric materials at different intensive or extensive quantities being constant.

**Figure 7.**(

**a**) Graph of isochoric molar entropy capacity versus absolute temperature; (

**b**) graph of molar temperature coefficient of energy ${\widehat{C}}_{V}$ versus absolute temperature. The graphs were calculated according to the Debye model for five different Debye temperatures on the examples given in Debye’s original work [11] and include low-temperature approximations (i.e., ${T}^{2}$ dependence in (

**a**) and ${T}^{3}$ dependence in (

**b**)).

**Table 1.**Energy forms in the context of caloric effects and related intensive and extensive quantities.

Caloric Effect | Energy Form | Conjugated Quantities | |||
---|---|---|---|---|---|

Intensive Quantity | Extensive Quantity | ||||

magnetocaloric | magnetisation energy | magnetic field | $\mathcal{H}$ | magnetisation | M |

elastocaloric | elastic energy | stress | $\sigma $ | strain | $\epsilon $ |

electrocaloric | polarisation energy | electrical field | $\u2107$ | polarisation | P |

barocaloric | compression energy | pressure | p | volume | V |

all | thermal energy ^{1} | temperature | T | entropy | S |

^{1}Thermal energy is also called “heat”.

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Feldhoff, A.
On the Thermal Capacity of Solids. *Entropy* **2022**, *24*, 479.
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Feldhoff A.
On the Thermal Capacity of Solids. *Entropy*. 2022; 24(4):479.
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**Chicago/Turabian Style**

Feldhoff, Armin.
2022. "On the Thermal Capacity of Solids" *Entropy* 24, no. 4: 479.
https://doi.org/10.3390/e24040479